MA/MSCMT-02
December Examination 2017
M.A. M.Sc. (Previous) Mathematics Examination
Real Analysis and Topology
Paper MA/MSCMT-02
Time 3 Hours Max. Marks 80
Note: The question paper is divided into three sections B and C.
Section A 8 × 2 16
(Very Short Answer Type Questions)
Note: Section contain 8 very short Answer Type Questions. Examinees have to attempt all questions. Each question is of 02 marks and maximum word limit is thirty words.
Define σ ring.
Define measurable function.
State Weierstrass approximation theorem.
State Minkowski's inequality.
Write the necessary and sufficient conditions for a bounded function f defined on the interval to be L-integrable.
Define Hilbert space.
231
MA/MSCMT-02 1900 3 (P.T.O.)
231
MA/ MSCMT-02 1900 3 (Contd.)
Define exterior of a set
(viii) Define compact topological space.
Section B 4 × 8 32
(Short Answer Type Questions)
Note: Section contain 08 short Answer Type Questions. Examinees
will have to answer any four question. Each question is of
08 marks. Examinees have to delimit each answer in maximum
200 words.
Let be a countable collection of sets of real numbers, then show
that.
m En m
n
n
n
o
If f is a bounded measurable function defined on a measurable set
then prove that f is L-integrable over E and
f(x)dx dx
E E

If a function is summable on then show that it is finite almost
everywhere on E.
Show that an orthonormal system
is complete iff it is closed.
State and prove Holder's inequality.
Prove that in a T2- space, a convergent sequence has a unique limit.
Show that the property of a space being a Haudorff space is a
hereditary property.
231
MA/ MSCMT-02 1900 3
Prove that every open continuous image of a locally compact space
is locally compact.
Section C 2 × 16 32
(Long Answer Type Questions)
Note: Section contain 04 Long Answer Type Questions. Examinees
will have to answer any two questions. Each question is of
16 marks. Examinees have to delimit each answer in maximum
500 words.
10) Show that every bounded measurable function f defined on a
measurable set E is L-integrable.
If the function f and g are Lebesgue integrable over the
measurable set E and if f g on then prove that
f(x)dx dx E E

11) Is A B A B Give reason in support of your answer.
Show that every metric space is a T2-space.
12) Prove that every interval is measurable.
13) State and prove Riesz-Fisher theorem.